Abstract.

We construct continuously parametrised families of conformally
invariant boundary operators on densities. These may also be viewed as
conformally covariant boundary operators on functions and generalise
to higher orders the first-order conformal Robin operator and an
analogous third-order operator of Chang-Qing. Our families include
operators of critical order on odd-dimensional boundaries. Combined
with the (conformal Laplacian power) GJMS operators, a suitable
selection of the boundary operators yields formally self-adjoint
elliptic conformal boundary problems. Working on a conformal manifold
with boundary, we show that the operators yield odd-order conformally
invariant fractional Laplacian pseudo-differential operators. To do
this, we use higher-order conformally invariant Dirichlet-to-Neumann
constructions. We also find and construct new curvature quantities
associated to our new operator families.
These have links to the Branson Q-curvature and include higher-order
generalisations of the mean curvature and the T-curvature of
Chang-Qing. In the case of the standard conformal hemisphere, the
boundary operator construction is particularly simple; the resulting
operators provide an elementary construction of families of symmetry
breaking intertwinors between the spherical principal series
representations of the conformal group of the equator, as studied by
Juhl and others. We use our constructions to shed
light on some conjectures of Juhl.

Key words and phrases:

ARG gratefully acknowledges
support from the Royal Society of New Zealand via Marsden Grants
13-UOA-018 and 16-UOA-051

LJP would like to thank the University of Auckland for its
hospitality on three separate occasions.

1. Introduction

Conformally invariant differential operators play a central role in
the global geometric analysis of Riemannian, pseudo-Riemannian, and
conformal structures
[14, 20, 55, 61]. The nature of
conformal geometry means that local issues can also be subtle and
difficult. For example, given a proposed leading symbol, a
corresponding conformally invariant differential operator may or may
not exist, depending on the conformal class. While important open
problems remain, nevertheless over the past decades there have been
significant advances in our understanding of these issues
[22, 33, 40, 41].

By comparison, relatively little is known about natural conformally
invariant differential operators along submanifolds. Here we treat an
aspect of this gap. Our work is primarily motivated by potential
applications to boundary value problems, conformally invariant
Dirichlet-to-Neumann constructions (cf. [5, 6, 7, 11, 12, 15, 23, 24]), and related issues
linked to geometric PDE, scattering, and the AdS/CFT correspondence of
physics [42, 52, 53, 54]. Although we do not treat continuous
families of non-local operators here, we believe that our results are
complementary to e.g. [7, 12, 15, 42] and that
the tools developed here should shed light on such families. The
results also give new constructions of families of representation
intertwining operators in the sense of Juhl’s families [47] and
the differential symmetry breaking operators of Kobayashi et
al. [50]. See also
[18, 28, 49].

In this paper, we construct continuously parametrised families of
conformally invariant differential boundary operators on densities.
We also construct curvatures associated to these operator families.
The new operators may be viewed as conformally covariant boundary
operators on functions and for most parameter values generalise to
higher orders the well-known first-order conformal Robin operator. We
show that some of the operators from the new families combine with the
GJMS operators of Graham et al. [41] to yield formally
self-adjoint elliptic conformally invariant higher-order Laplacian
boundary problems. In conjunction with these problems, we construct
higher-order conformally invariant Dirichlet-to-Neumann operators that
provide realisations of odd-order conformally invariant fractional
Laplacian pseudo-differential operators. In the later sections, we
work in the setting of general conformally curved manifolds with
boundary, but by virtue of the conformal invariance, the results may
be applied to conformally compact manifolds and in particular to
asymptotically hyperbolic and Poincaré-Einstein manifolds. Earlier
work of Branson and the first author in [6] used tractors
to construct new families of boundary operators. (We will review
tractors in Section 3.2, below.) In the present article,
the approach is again via tractors, but it is simpler, quite
different, and most importantly, more effective; we close gaps in
[6].

We now explain in more detail the problems treated in this paper, the
motivations for these problems, and the results obtained. Throughout
our work, M is a manifold of dimension n, c is a conformal
equivalence class of metrics on M, and \textslg is a metric in c.
The term hypersurface will always refer to a smooth conformally
embedded (n−1)-dimensional conformal manifold in (M,c).
In this context, denotes the conformal equivalence class of the
metric on Σ induced by any \textslg∈c, and will denote the
metric on Σ induced by such a \textslg. We will often assume that M
is initially given as a manifold with boundary, and Σ=∂M
is the boundary of M. But in this case, for the convenience of
treating the local issues, we will assume that we have extended M to
a collar neighbourhood of Σ (so that Σ consists of interior
points of M). Unless we indicate otherwise, we will always assume
that n≥3. For simplicity of discussion, all structures we
consider below will be taken to be smooth to infinite order, and we
shall restrict to the case of Riemannian signature. These restrictions
can be relaxed to a large extent with little change.

The operators in our new families are natural. In
Section 2, below, we will discuss the idea of natural
differential operators along a hypersurface, as well as the idea of
scalar Riemannian hypersurface invariants. We note here, however,
that a natural differential operator along a hypersurface is a rule
that defines a differential operator on a neighbourhood of any
hypersurface in any ambient conformal n-manifold (M,c).
In a similar way, a hypersurface invariant is actually a rule which
defines an invariant in a neigbourhood of any hypersurface
of any (M,c). (In both cases, however, we may require
and (M,c) to meet certain conditions.) The operators in our new
families are parametrised by n (in each case in some infinite set
\sf D⊆Z≥3) and a real number w. We
say that w is a weight. The operators in a given family
will always be determined by a single universal symbolic formula, so
we will
sometimes use the word “operator” to refer to any of our new
families of operators or to any other family of operators determined
by a single universal symbolic formula. Similar remarks apply to
families of hypersurface invariants. Although our main results give
operators that, along the hypersurface, are determined by the
conformal embedding, the development of these results involves
operators that can depend on a choice of metric \textslg within c.

Some of the operators and curvature quantities we construct in this
paper are motivated by certain well-known natural differential
operators and their associated curvature quantities. One such
operator and curvature pair is the conformal Robin operator
(of Cherrier [17]) and the mean curvature. To define the
conformal Robin operator, we begin by identifying Σ with its image
submanifold under an embedding ι:Σ→M. Fixing
\textslg∈c determines a metric , and we write na for the
associated unit conormal field to Σ. This conormal field
together with the metric determines a unit normal vector field
na=\textslgabnb. We let H (or sometimes H\textslg) denote the mean
curvature of Σ. Let ∇ denote the Levi-Civita connection
determined by \textslg, and let C∞(M) and C∞(Σ)
denote the sets of all smooth real-valued functions on M and Σ,
respectively. The conformal Robin operator is the operator δ:C∞(M)→C∞(Σ) defined by the composition of
f↦na∇af−wHf with restriction to
C∞(Σ). Here w is a real parameter. As before, we say
that w is a weight. We define conformal covariance and
bidegree in Definition 2.1 below. For all w∈R,
δ is conformally covariant of bidegree (−w,1−w).

The operator δ is a Robin operator because it mixes Dirichlet
and Neumann data. Its conformal covariance is important for forming
well-posed conformal boundary problems involving the conformal
Laplacian (or “Yamabe operator”) [6, 17, 23, 45] on
the n-dimensional interior. Let w=1−n/2, which is the weight
selected by the covariance properties of the Yamabe operator, and
consider the limiting case in which n=2. Then w=0, and δ
is just the Neumann operator na∇a. The mean curvature H
drops out of the formula for δ, but it retains an interesting
limiting link with δ: In this specific case, H is the
geodesic curvature and transforms conformally by the rule
. Here
and throughout this paper, Υ∈C∞(M), and
ˆ\textslg=e2Υ\textslg.

In connection with the problem of understanding Polyakov-type formulae
for conformally covariant elliptic differential operators on compact
4-manifolds with boundary, Chang and Qing discovered a third-order
analogue, P\tiny\it CQ3, of the conformal Robin operator. (They denoted
this operator by P3.) See [16]. This operator acts along the
boundary of the compact 4-manifold. Chang and Qing showed that
P\tiny\it CQ3 is naturally associated to a scalar curvature quantity T=TCQ
along the boundary and that the transformation rule for T under
conformal change of metric is . In this and other ways (as explained in
Section 7, below) the relation between P\tiny\it CQ3 and TCQ
is analogous to the relation between the Neumann operator and the geodesic
curvature on a surface with boundary which we described above. By
[16], P\tiny\it CQ3 is conformally covariant of bidegree (0,3).

In [41], Graham et al. defined a conformally invariant kth
power of the Laplacian on a conformal n-manifold (M,c). This
operator is commonly known as the GJMS operator of order
2k, and we will denote it by P2k. It acts on conformal
densities, and it is well-defined for all positive integers k within
an appropriate range. (We will discuss conformal densities and their
weights in Section 3.2, below.) In [4], Branson
used Pn to define a scalar curvature quantity Qn on a compact
Riemannian manifold (M,\textslg) without boundary of even dimension n≥4. This curvature is called the Q-curvature of (M,\textslg). The
operator Pn may be viewed as a conformally covariant operator
P\textslgn:C∞(M)→C∞(M) of bidegree (0,n).
(See Proposition 3.2.) Under conformal change of
metric, Qn transforms according to the rule
Qˆ\textslgn=e−nΥ(Q\textslgn+P\textslgnΥ).
The pair (δ,H) in dimension n=2 and the Chang-Qing pair
(P\tiny\it CQ3,TCQ) in dimension n=4 give, in a certain sense, odd-order
boundary analogues of the pair (Pn,Qn). This has generated
considerable interest ([11, 13, 47, 53, 54, 59])
and motivates Problem 1.2, below.

The statement of Problem 1.2 involves the notion of the
transverse order of a boundary operator along Σ, as
defined in Definition 1.1, below.
Definition 1.1 is adapted from the definition of
the normal order of a boundary operator given in
[6]. Our definition uses a defining function for Σ on
a neighbourhood U of a point p∈Σ. Here and below, a such a
function is a function t∈C∞(U) such that Σ∩U is
the zero locus of t and dt is nonvanishing on U.

Definition 1.1.

Let a Riemannian manifold (M,\textslg), a
hypersurface in (M,\textslg), a vector bundle F over M,
and a differential operator B:F↦F|Σ be given. Also
let p∈Σ be given, and suppose that t is any defining function
for Σ on some neighbourhood U of p. For any
m∈Z>0, we say that B has transverse orderm at p if there is a smooth section V of F such that
B(tmV)|p≠0 but B(tm+1V′)|p=0 for all smooth
sections V′ of F. If B(tV)|p=0 for all smooth sections
V of F, then we say that B has transverse order 0 at p.
Now let m∈Z≥0 be given. If B has transverse
order m at every p∈Σ, then we say that B has transverse
order m. Finally, suppose that m>0. If B has transverse order
less than m at every p∈Σ, then we say that B has transverse
order less thanm or transverse order at mostm−1.

In the definition here, and also throughout the article, we interpret
notation to mean vector bundles or their smooth section spaces
according to context. The properties described in this definition are
independent of the choice of the defining function t. The
transverse order at a point p∈Σ measures the number of
derivatives in directions transverse to Σ at p. If we say that
a natural hypersurface differential operator B has order m, order
less than m, transverse order m, or transverse order less than
m, we mean that the property holds for all hypersurfaces in all
possible Riemannian manifolds; for the operator families that we deal
with in this paper, the presence of these properties may depend on the
values of the parameters w and n mentioned above. The conformal
Robin operator has transverse order 1 in all dimensions, and P\tiny\it CQ3
has transverse order 3.

In the statement of Problem 1.2 that follows, and throughout this
paper, differential operators will act on everything to their right,
unless parentheses indicate otherwise. In addition, we will always
indicate multiplication operators by juxtaposition. For example, let
differential operators Op1,Op2:C∞(M)→C∞(M)
and functions f1,f2∈C∞(M) be given. Then
Op1f1Op2f2 denotes Op1 acting on the product of f1 and
Op2f2. All differential operators in this paper will be linear.

We may now state one of the main problems of this paper.

Problem 1.2.

For some K∈Z>0, construct a family of natural
hypersurface operators P\textslgw,K:C∞(M)→C∞(Σ) parametrised by the dimension n and a real
parameter w such that the following properties hold: (1) For all
w∈R, P\textslgw,K has order at most K. There is a
small finite set E such that for all w∈R∖E,
P\textslgw,K has order and transverse order K. The set E
may depend on n.
(2) For all f∈C∞(M) and all
w∈R,
Pˆ\textslgw,Kf=e−(K−w)ΥP\textslgw,Ke−wΥf.
(3) The family P\textslgw,K determines a scalar curvature
quantity Q along Σ with the property that. This scalar curvature quantity is a local scalar Riemannian
hypersurface invariant.

One of our main objectives will be to find solutions to
Problem 1.2 in which the set E of exceptional weights
is as small as possible or solutions in which some specific
undesirable weight (such as w=0) is absent from E.

We will be especially interested in higher-order generalisations of
the conformal (Cherrier-)Robin operator and the Chang-Qing operator.
The Chang-Qing operator has order and transverse order n−1=3, the
dimension of Σ. Similarly, in the limiting case in which n=2,
the conformal Robin operator likewise has order and transverse order
n−1. In general even dimensions n, establishing the existence of
conformal boundary operators of order and transverse order n−1 is
rather delicate. Such operators sit in a special position that is in
part analogous to the place of the dimension-order GJMS operators of
[41]. Thus, in the setting of Problem 1.2, if n
is even, we will say that a boundary operator of order and transverse
order n−1 is a critical operator and that n−1 is the
critical order. Paramount in our considerations here is
finding a solution to Problem 1.2 that includes such
operators as part of a continuous family. For emphasis we state the
following special case of Problem 1.2:

Problem 1.3.

In Problem 1.2, suppose that n is even and
K=n−1. Find a solution such that P\textslg0,K is a critical
operator.

In this paper, we construct families of operators which solve
Problem 1.3 in general even dimensions n≥4. We do this
by first constructing three families of operators which solve
Problem 1.2. Let K be as in Problem 1.2.
Then for all K∈Z>0,
the operator family δ0K of Theorem 5.12 solves
Problem 1.2 in the simplest setting, namely the case of
conformally flat metrics. (Throughout this paper, conformally flat
means what is sometimes referred to as “locally conformally flat”,
that is, the Weyl and
Cotton tensors both vanish. Also, note that we suppress the \textslg and
the w from the operator notation.) The δ0K family is
defined in all dimensions n≥3. Next, for all
K∈Z>0, the operators δJ,k of Theorem 5.16,
with J+k=K, are defined for general Riemannian conformal structures
in suitable dimensions and for conformally flat structures in all
dimensions n≥3. Similarly, for all K∈Z>0, the
operators δK of Lemma 5.4 are defined on
general Riemannian conformal structures in all dimensions n≥3.
The families δJ,k and δK solve Problem 1.2.

The δ0K operator family solves Problem 1.2 on
the standard conformal hemisphere, and in this case, the operators
δ0K provide families of symmetry breaking intertwinors
between the spherical principal series representations of the
conformal group of the equator, as studied by Juhl and others
[18, 47, 50]. There is some appeal in this picture,
as here the symmetry breaking (the reduction of the conformal group of
the sphere to a subgroup preserving the closed hemisphere) is
manifestly governed
by the normal tractor. By using the tools in, for example,
[34], one may develop generalisations of the
δ0K and δJ,k families which treat differential forms
and other tensors. Such generalisations should be of some interest to
the more general intertwinor programme as in
[28, 48], but we do not investigate this idea in
this paper.

As part of a construction of a family of conformally invariant
higher-order Dirichlet-to-Neumann operators, Branson and the first
author discover and construct a family δBGK of conformally
invariant hypersurface operators in Theorem 5.1 of
[6]. Although used at a discrete set of weights in
[6], by construction the operators are available on any
density bundle, and in the interesting work [47], Juhl studies
the resulting continuously parametrised families of boundary operators
(as well as so-called “residue families”) and considers some
problems linked to those studied here; see for example [47, Section
1.10 and Section 6.21]. Unfortunately, as evident in
[6, Theorem 5.1] and its proof, the δBGK
family does not provide an answer to Problem 1.3. In
[43], Grant constructs a modification δG3 of
δBG3 which solves Problem 1.3 in the dimension n=4
case. (See also Stafford [57] and Case [11].)
However, Grant’s work was specific to third-order operators, and the
problem of finding higher-dimensional analogues of the Chang-Qing pair
(P\tiny\it CQ3,TCQ) has, to our knowledge, remained open up to now.

For every even integer n≥4, the δ0K family includes
an operator which solves Problem 1.3 for conformally flat
Riemannian conformal manifolds of dimension n, and the δJ,k
family contains a solution to this same problem for general Riemannian
conformal manifolds of dimension n. See Theorem 7.10,
below.

In Section 6.1, the operators δJ,k and δK are
used to set up conformally invariant elliptic boundary problems
for the GJMS operators and then to construct non-local operators. The
first main result is Theorem 6.3, which, for any GJMS
operator P2k, describes a corresponding conformally invariant
boundary system B such that the boundary value problem (P2k,B)
is formally self-adjoint. The boundary system B is given by
(46). Lemma 6.5 then shows that the boundary
problem (P2k,B) satisfies the Lopatinski-Shapiro
condition. The pair (P2k,B) is properly elliptic, and in
Theorem 6.7, we use this fact, together with the other
properties of the pair, to construct Dirichet-to-Neumann operators.
The statement of Theorem 6.7 is too technical to be given in
this introduction, but we may easily state two corollaries of the
theorem here. In these corollaries, ¯E[w] denotes the bundle of
conformal densities of weight w associated to ; here w
is any given real number. The first corollary follows from the case
k=n/2 (n even)
of Theorem 6.7 and Lemma 6.1.

Corollary 1.4.

Let a compact Riemannian conformal manifold with boundary
(M,Σ,c) of even dimension n≥4 be given. Let B be as
in (46), and suppose that (Pn,B) has trivial kernel. Then
for 2m=1, 3, 5, …, n−1, Theorem 6.7 yields
conformally invariant non-local operators

which have leading term .

Corollary 1.4
produces
a critical-order (i.e. order n−1) fraction Laplacian as the
case 2m=n−1.
Such an operator was missing from the results of [6].

Corollary 1.5.

Let a compact Riemannian conformal manifold with boundary
(M,Σ,c) of dimension n≥3 be given. Also let
k∈Z>0 be given, and suppose that (1) n is odd, or
(2) n is even and k≤n/2, or (3) (M,c) is conformally flat.
Finally, let B be as in (46), and suppose that (P2k,B)
has trivial kernel. Then for 2m=1, 3, 5, …, 2k−1,
Theorem 6.7 yields conformally invariant non-local operators

PT,k2m:¯E[m−n−12]→¯E[−m−n−12]

which have leading term .

For each solution P\textslgw,K to Problem 1.2, we will
say that the associated scalar curvature quantity Q is a Q-type curvature. This Q-type curvature has certain properties
analogous to those of Branson’s Q-curvature.
However, for any dimension n and any K∈Z>0, if a
solution P\textslg0,K has transverse order K, then we will say
that the Q-type curvature associated to P\textslgw,K is a
T-curvature of order K and that (P\textslg0,K,Q\textslg)
is a T-curvature pair of this order. We often let T\textslg
denote Q\textslg in this case. In Section 7, we will see that
for any given even dimension n≥4, there are T-curvature pairs
of all orders. Specifically, we will establish the following theorem:

Theorem 1.6.

Let n0∈Z≥4 be given, and suppose that n0 is
even. Then in dimension n=n0, there are canonical T-curvature
pairs

Let \textslg∈c be given. Proposition 7.15, below, gives
conditions which ensure that for all m∈Z≥2, there
is a metric ˆ\textslg∈c which induces and which satisfies
along Σ. Under
these same conditions, ; this will become
clear later. So under these conditions, the hypersurface Σ is
minimal for the metric ˆ\textslg in the sense that Hˆ\textslg=0 along
Σ, and Σ also satisfies the related higher-order condition
Tˆ\textslg2=⋯=Tˆ\textslgm=0 along Σ.

We have computed explicit symbolic symbolic formulae for our
new operator families and their Q-type curvatures in certain cases.
One such formula is the following formula for the Q-type curvature
of the δ1,2 family:

(1)

3na∇a\sf J−(n−2)nanbnc∇a\sf Pbc+6H\sf J−6(n−2)Hnanb\sf Pab+2(n−2)H3.

Here \sf Pab is the Schouten tensor, and \sf J=\sf Paa. (We
define the Schouten tensor in Section 3, below.) We give
explicit formulae for δ1,2 and a few of our other operator
families and their curvatures in Section 8, below. Some of
these formulae are valid only in certain dimensions, as explained in
that section.

Our operator and curvature constructions use the Fefferman-Graham
ambient metric of [25, 26], its link to tractors[9], and the tractor construction of the GJMS operators
developed in [35]. We will often work with symbolic formulae
which are polynomial in the parameter w of Problem 1.2
and rational in the dimension n. As a consequence, our proofs will
often use polynomial continuation in w and rational continuation in
n.
To treat these notions, we develop some tools and results in
Section 4, below.

2. Conformally covariant operators along a
hypersurface

In this section, we work with arbitrary
Riemannian metrics. We will generally employ Penrose’s abstract index
notation. (If no ambiguity will occur, however, we will sometimes
omit indices from tensors.) We shall write Ea to denote the
space of sections of the tangent bundle TM over M, and Ea for
the space of sections of the cotangent bundle T∗M. (In fact, we
will often use the same symbols for the bundles themselves.) We write
E for the space of real-valued functions on M (or for the
trivial bundle M×R).
All functions, vector bundles, and sections of vector bundles will be
assumed to be smooth, meaning C∞. An index which appears
twice, once raised and once lowered, indicates a contraction. The
metric \textslgab and its inverse \textslgab enable the identification
of Ea and Ea, and we indicate this by raising and lowering
indices in the usual way. In Section 3, below, we will
discuss the Riemannian and Weyl curvature tensors and the Ricci and
Schouten tensors; throughout much of
this paper, the term “Riemannian curvature tensor” will include
these tensors and the traces of the Ricci and Schouten tensors.

Let an embedded hypersurface Σ of M be given, and suppose that
M and Σ are both orientable. We will usually work locally and
assume that Σ is the zero locus Z(t) of a defining
function t, so the orientability assumptions will usually not
impose any restriction. We will need to consider geometric quantities
determined on Σ. Rather than deal with the awkwardness of fields
and quantities which are defined only along Σ, we will define
extensions of these into a neighbourhood of Σ; we emphasise that
our final results will not depend on the choice of these extensions.
We will calculate in a neighbourhood on which dt is nowhere zero,
and we define

(2)

na:=dt|dt|\textslg

in this neighbourhood. The level sets of t define a foliation
Σt, with Σ=Σ0, such that na gives the unit
conormal field along each leaf t=constant.

For operators on functions, the notion of naturality along a
hypersurface is an obvious adaptation of the usual notion from
Riemannian geometry (see, e.g., [2, 58]). A
natural differential operator along a hypersurface is a
differential operator which, in a neighbourhood of the hypersurface,
may be expressed by a universal symbolic formula which is polynomial
in the Levi-Civita connection ∇ of (M,\textslg) and has tensor-valued
pre-invariants as coefficients. We refer the reader to
Section 2.4 of [37] for the definitions of scalar- and
tensor-valued Riemannian hypersurface pre-invariants and
invariants. In this paper, any family of natural operators
will always be given by the same universal symbolic formula for all
possible conformal manifolds (M,c); a similar remark applies to
families of invariants. The symbolic formula for a family of such
operators or invariants will be a polynomial in the conormal field
na, the Riemannian metric \textslgab, its inverse \textslgab, the
Riemannian curvature tensor Rabcd, the mean curvature
H, and the Levi-Civita connection ∇a. (We give examples of
symbolic formulae for operators and invariants at various points in
this paper. Tractors, the trace-free part of the second fundamental
form, and various operators will appear in some of these formulae, but
it will always be possible to expand these formulae and write them as
polynomials of the above type.) The coefficients in the polynomial
formula for a family of operators or invariants will be real functions
of the dimension n and the weight parameter w that we discussed in
Section 1, above. This real function will be polynomial
in w and rational in n. In the case of an invariant, w will be
absent from the formula. For most of the universal formulae that we
work with in this paper, we may assume that ∇a never explicitly
hits nb or H.

In Section 3.2, below, we will incorporate conformal
densities into our operators and invariants. The above definitions of
natural operators and hypersurface invariants will extend to this
situation in the obvious way. Our work will also require the notion
of natural differential operators between sections of tractor bundles.
As we will see in Section 3.2, below, a tractor bundle is a
finite-dimensional vector bundle over (M,c). We
will see that for each g∈c, a tractor bundle decomposes into a
direct sum of tensor bundles. Naturality of a differential operator
between sections of tractor bundles will mean naturality with respect
to any such decomposition. Our earlier discussion concerning
families of operators and invariants extends to our work with
densities and tractors in an obvious way.

Now suppose that we choose an orientation for Σ. Then on Σ,
hypersurface invariants and natural differential operators along Σ
are independent of the choice of the defining function t, because
we insist that dt/|dt|\textslg be consistent with the
orientation. Such invariants and operators need not be uniquely
determined off of Σ. Let F denote E or the set
of all smooth sections of any power of the tractor bundle over M,
and let V∈F be given. Also let a natural hypersurface
differential operator Op:F→F|Σ be given. Then
OpV will usually denote the restriction of OpV to Σ, but
we will sometimes indicate the restriction explicitly by writing
OpV|Σ. We will sometimes write Op:F→F
instead of Op:F→F|Σ.

One example of a scalar hypersurface invariant is the mean curvature
. Here Lab is the second fundamental
form, and is the hypersurface metric determined by \textslg as
in Section 3.1, below. Similarly, Lab is a tensor-valued
hypersurface invariant, and is another scalar invariant.

For any w∈R, a scalar Riemannian hypersurface invariant K
determines a conformal invariant of weight w if it
satisfies the conformal covariance relation
K(e2Υ\textslg,t)=ewΥΣK(\textslg,t)
for all Υ∈E. Here ΥΣ is the pullback of
Υ to the hypersurface Σ. Thus K determines a
homogeneous function on Q, the bundle of conformal metrics. This
function represents an invariant conformal density of weight w.

Definition 2.1.

We say that a natural differential operator P\textslg:E→E
is conformally covariant of
bidegree(w1,w2) in R2 if
Pˆ\textslgV=e−w2ΥP\textslgew1ΥV
for all \textslg and for all Υ and V in E.
This definition extends to hypersurface operators and operators on
tractors in the obvious way.

In Section 3.3, we will replace conformal covariance with the
equivalent notion of conformal invariance. This will simplify the
discussion and calculations. Our aim will be to construct special
natural conformally invariant operators; there will be no attempt to
classify operators. The strategy is to build these in such a way that,
by construction, they satisfy the naturality conditions. Achieving
conformal invariance, although more subtle, will eventually be seen to
yield to the same approach.

Before continuing, we consider two examples of conformally covariant
natural operators. For the conformal Robin operator
δ:E→E|Σ, naturality is evident from the
formula

(3)

δf:=na∇af−wHf.

We will also let δ1 or δ1,\textslg,w denote the conformal
Robin operator.
In any dimension n≥2, the mean
curvature H satisfies the conformal transformation rule
.
From this it follows that for any w∈R, the conformal Robin
operator is conformally covariant of bidegree (−w,1−w).

A second-order analogue of δ1 is the operator δ2 (also
denoted by δ2,\textslg,w) given by the formula

Here Δ=∇i∇i and f∈E. In Section 3, below,
we will see that \sf Pab and J are Riemannian invariants. From
this it will follow that δ2 is manifestly natural, since each
object in the formula for δ2 is determined by the data of the
ambient Riemannian structure on M and the embedding. No other
information is involved. The normal vector field na appears in
the formula, but along Σ, the value of δ2f is independent
of the extension of na off of Σ. Consideration of the
tractor formula for δ2 in Section 5.1 will show that if
w is any real number, then δ2,\textslg,w is conformally
covariant of bidegree (−w,−w+2). If w=1−n/2, then
δ2=−□, where □ is as defined in (15), below. In
this case, □ is the Yamabe operator for (M,c). In
Section 8.1, we will relate δ2 to the
intrinsic Yamabe operator on Σ and to a second-order hypersurface
operator of [43]. Finally, note that δ2 is
considerably more complex than δ1. It is easily seen that
there is exponential growth in complexity as order increases. A
naïve approach to conformal submanifolds will therefore not
suffice.

3. Conformal geometry and hypersurfaces

Let ∇a denote the Levi-Civita connection on a Riemannian
manifold (M,\textslg) of dimension n≥2.
The Riemannian curvature
tensor R on (M,\textslg) is defined by

R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z.

Here X, Y, and Z are arbitrary vector fields. In abstract index
notation, R is denoted by Rabcd, and R(X,Y)Z is
XaYbZdRabcd. In dimensions n≥3 this can be
decomposed into the totally trace-free Weyl curvatureCabcd
and the symmetric Schouten tensor\sf Pab according to

(4)

Rabcd=Cabcd+2\textslgc[a\sf Pb]d+2\textslgd[b\sf Pa]c,

where [⋯] indicates antisymmetrisation over the enclosed
indices. In (4), \sf Pab is a trace modification of the
Ricci tensor given by

(5)

Here
Rcab=Rcacb, and \sf J=\sf Paa. In dimensions 2 and
3 the Weyl tensor Cabcd vanishes identically by dint of its
symmetries. In dimension 2, (5) does not define a
Schouten tensor in the sense we require in this work. By adding
additional structure (a Möbius structure [8]),
however, one may define a Schouten tensor in dimension 2 in such a way
that that it has conformal properties similar to the Schouten tensor
in higher dimensions.

3.1. Riemannian hypersurfaces

In this subsection, we discuss covariant derivatives and various
structures on Σ and relate them to covariant derivatives and
structures on M. These ideas will facilitate the understanding of
some of the example formulae in Section 8, below. This
material is standard and appears in such sources as [56],
Chapter 1, but we develop it here to set notation fit with our current
approach via defining functions, cf. [19, 37, 43, 57].

To begin, let Σ denote a hypersurface in (M,\textslg) as in
Section 1, and let t denote any local defining
function for Σ, as in Section 2. We are concerned with
local theory here, so without loss of generality, we may assume that
dt is nowhere zero. Let na be as in (2). Recall
that the level sets of t define a foliation Σt, with
Σ=Σ0. For each leaf t=t0, the embedding ι:Σt0→M induces an injective bundle map ι∗:TΣt0→TM, and we shall simply identify TΣt0
with its image. Dually, T∗Σt0 is naturally a quotient of
T∗M. The bundle epimorphism T∗M→T∗Σt0 is split by
the metric, and so T∗Σt0 is naturally identified with the
subbundle of T∗M consisting of 1-forms orthogonal to the conormal
bundle to Σt0. We shall thus use the same abstract indices
for TΣt0 as we do for TM. We will, however, use a bar to
denote objects intrinsic to Σ or to any leaf of the foliation.
Thus ¯Ea (resp. ¯Ea) will denote the subbundle of Ea
defined by the foliation (resp. the subbundle of 1-forms annihilating
na), the spaces of smooth sections of these, or the restrictions of
these objects to a leaf Σt0 (for any t0∈R
where the foliation is defined). For example, ua∈Ea is a
section of ¯Ea if and only if uana=0. In particular, this
applies along Σ=Σ0, where our interest really lies, and any
section of ¯Ea (resp. ¯Ea) along Σ will be assumed to
be the restriction to Σ of a section of ¯Ea (resp. ¯Ea)
defined in a neighbourhood of Σ. This idea will be extended to
tensor powers in an obvious way with little further mention.

Next, let Πab:=δab−nanb. As a section of End(TM), this defines projections TM→TΣt0 and T∗M→T∗Σt0 in the obvious way. Thus, for example, the formula
defines a symmetric 2-tensor that restricts
to give an induced intrinsic metric along any
Σt0. Similarly , which is
consistent with raising indices using the ambient metric \textslgab. We
write for the corresponding intrinsic Levi-Civita connection
along the leaves, and R denotes the corresponding Riemannian
curvature. Although we shall be finally interested in these quantities
only along Σ=Σ0, it is convenient to have fixed an extension
off Σ in this way; this is consistent with our treatment of vector
fields and tensors in general, as discussed above. They depend
smoothly on points in the foliated neighbourhood of Σ.

Along Σ, and indeed along each leaf Σt0, the second
fundamental form of the embedding, Lab, is given by the formula
Lab=Πac∇cnb. (Note that the sign of L here
differs from that of many sources, including [56].) By the
Weingarten equations Lab is
symmetric. An easy exercise shows that naLab=0, so L defines
a smooth section of ¯E(ab), the second symmetric power of
¯Ea. Let Va,Tb∈¯Ea be given. Then
LabVaTb=−\textslg(na,∇VT) along Σ, again by the
Weingarten equations. Thus if dt is compatible with a given
orientation on Σ, then Lab, and hence also the mean
curvature H, are independent of the choice of the defining
function t.
The metric trace-free part of L, denoted by , is given
by .

An easy computation shows that H=(1/(n−1))∇ana.
Hypotheses 4.1, below, will refer to a modified mean
curvature tensor G given by G=∇ana. The purpose of
defining G in this way is to obtain a curvature given by a
symbolic formula which involves ∇a and na but not the
dimension n.

It is easily verified that is
related to the ambient Levi-Civita connection ∇ by the Gauss formula

(6)

where Va∈¯Ea; in particular
. From this follows the
classical Gauss equation, which we give in our current notation:

Proposition 3.1.

Let denote the intrinsic Riemannian curvature
tensor. Then

3.2. Conformal structures

A conformal geometry is a manifold of dimension at least 2 equipped
with a conformal structure, i.e. an equivalence class c of
Riemannian metrics such that if \textslg,ˆ\textslg∈c, then
ˆ\textslg=e2Υ\textslg for some Υ∈E. Our operator constructions
will require several results and techniques from conformal geometry.
For further details see [3, 9, 35], or [19] for
a recent overview.

We begin by interpreting a conformal structure c as a ray
subbundle Q⊆S2T∗M whose fibre over x∈M consists
of the set of all metrics at x which are conformally related to some
given metric \textslg at the point x. The principal bundle π:Q→M has structure group R>0, so for any w∈R, the
representation R>0∋x↦x−w/2∈End(R) induces a natural (oriented) line bundle on M that we term the
bundle of conformal densities of weight w. We let E[w]
denote the space of sections of this bundle or the bundle itself.
There is a tautological section {g}ab of S2T∗M⊗E[2]
that is termed the conformal metric. Similarly, {g}−1 is a
section {g}ab of S2TM⊗E[−2]. Henceforth, {g}ab
and {g}ab will be the default objects that will be used to
identify TM with T∗M⊗E[2] and to raise and lower indices
associated to these bundles (even when a metric \textslg∈c has been
chosen). As a consequence, tensors, hypersurface invariants, and
natural differential operators will now carry weights. For example,
if Rabcd is the Riemannian curvature tensor associated to
a metric \textslg∈c, then Rabcd will have weight 0.
Similarly, Rabcd will denote {g}ceRabed, and this
tensor will have weight 2. The weights of other such curvature
tensors will be evident from their definitions. We will assign
weights to na, Lab, H, and in
Section 3.3. If P is a natural differential operator
acting between densities, then for some c∈R and all
conformal manifolds (M,c) (or all conformally flat conformal
manifolds (M,c)) of appropriate dimensions, P will map E[w] to
E[w−c]. For example, for any f∈E[w], we have Δf={g}ab∇a∇bf∈E[w−2].

A metric \textslg∈c is equivalent to a positive section ξ\textslg of
E[1] via the relation \textslgab=(ξ\textslg)−2%gab. We say that
ξ\textslg is the scale density associated to \textslg. Let
w∈R and a section σ of E[w] be given. We may
write σ=(ξ\textslg)wf for some f∈E. It is easily verified
that the Levi-Civita connection ∇, of \textslg, acting on E[w], is
the connection mapping σ to (ξ\textslg)wdf, where d is the
exterior derivative. Let Eb[w]=Eb⊗E[w], and let
ˆ∇ denote the Levi-Civita connection associated to the
metric ˆ\textslg=e2Υ\textslg. Then for all μb∈Eb[w],

(7)

ˆ∇aμb=∇aμb+(w−1)Υaμb−Υbμa+{g}abΥcμc,

where Υa=∇aΥ.

On a general conformal manifold there is no distinguished connection
on TM. There is, however, a canonical conformally invariant
connection on a slightly larger bundle, and this is called the
(conformal) tractor connection[3]. It is linked, and
equivalent to, the normal conformal Cartan connection of Elie Cartan
[10]. On a conformal manifold (M,c) of dimension n≥3,
let T (or TA as the abstract index notation) denote the
(standard) tractor bundle. This bundle is a canonical rank n+2
vector bundle equipped with the canonical (normal) tractor connection
∇T. This connection is conformally invariant. We usually
write ∇ instead of ∇T. We let TΦ denote any
tensor power of T, including E. To distinguish different (or
potentially different) powers of T, we write TΦ1 and
TΦ2. Let TΦ[w]=TΦ⊗E[w]. Also let T,
TΦ and TΦ[w] denote the spaces of sections of these
bundles. A choice of metric \textslg∈c determines a splitting of
T, i.e. an isomorphism

(8)

T\textslg≅E[1]⊕Eb[1]⊕E[−1].

We may write T\textslg=(σ,μb,ρ) to indicate that T is an
invariant section of T, and (σ,μb,ρ) is its image
under the splitting given by (8). In general, a conformally
related metric ˆ\textslg
determines a different splitting of
T. If T\textslg=(σ,μb,ρ), then

(9)

Tˆ\textslg=(σ,μb+σΥb,ρ−{g}cdΥcμd−12σ{g}cdΥcΥd).

To facilitate our computations, we introduce three algebraic splitting
operators,

YA∈EA[−1],ZAb∈TAb[−1]:=TA⊗Eb⊗E[−1],XA∈EA[1],

which administer the isomorphism (8) determined by the metric
\textslg∈c. If TA\textslg=(σ,μb,ρ), then TA=σYA+μbZAb+ρXA. Note that (9) determines the
transformations of YA, ZAb, and XA under conformal change
of metric. These are easily computed and are given explicitly in
[35].

There is a conformally invariant tractor metrich on T
which is preserved by ∇T. We let h# denote the
co-metric associated to h on the dual bundle to T. In terms
of the splitting operators, the tractor metric is given by
Figure 1.

YAZAcXAYA001ZAb0{g}bc0XA100

Figure 1. The tractor metric

In a symbolic tractor expression with tractor indices, one may
eliminate all references to h and h# as follows. First, if
one index of h or h# is contracted with an index of some
other tractor, one may eliminate the reference to h or h# by
raising or lowering the index of this other tractor. On the other
hand, if h or h# has only free indices, then one may express
h and h# in terms of the splitting operators X, Y, and
Z. For example, hAB=ZAcZBc+XAYB+YAXB, as noted in
[35].

The tractor connection is usefully encoded in the formulae for the
tractor Levi-Civita coupled derivatives of the splitting operators:

We will also use the conformally invariant tractor W-curvature
WABCE as described in e.g. [35]:

(13)

WABCE=(n−4)ZAaZBbΩabCE−2X[AZB]b∇pΩpbCE.

From this formula and well known results it follows easily that c
is conformally flat if and only if WABCE=0.

The notation ψ:TΦ1[w]→TΦ2[w−c]|Σ or
ψ:TΦ1[w]→TΦ2[w−c] will indicate that
ψ is a family of natural differential operators parametrised by
w and n. Here c is a real constant and w is a real
number.
The family ψ is a rule given by a universal symbolic formula.
This rule defines an operator for all (M,c) of dimension n, all
bundles TΦ1, and all w∈R. (We may require (M,c) to
meet various conditions, and TΦ2 will depend on TΦ1
and ψ.) If w appears in the symbolic formula for ψ, we
set w equal to the weight of the bundle on which ψ acts, unless
we explicitly indicate otherwise. Since ψ is given by a
universal symbolic formula, we will sometimes refer to ψ as an
“operator” rather than a family of operators.

One important family of conformally invariant natural operators on
weighted tractors is the family
D:TΦ[w]→TA⊗TΦ[w−1] defined as
follows:

(14)

DAV=w(n+2w−2)YAV+(n+2w−2)ZAb∇bV−XA(Δ+w\sf J)V.

Cf. [3]. For developments of D and proofs of its conformal
invariance, see [21, 30]. Another important family of
natural operators is the family
□:TΦ[w]→TΦ[w−2] given by

(15)

□V=(Δ+w\sf J)V.

If w=1−n/2, then □ is the Yamabe operator, which is conformally
invariant. We will use D and □ in our main operator
constructions.

3.3. Conformal hypersurfaces

Let a hypersurface ι:Σ→M be given. In this subsection, we
present the necessary elements of basic conformal hypersurface
geometry.

Let \textslg,ˆ\textslg∈c be given. These metrics induce conformally
related metrics and on Σ, and so c induces a
conformal structure on Σ. We term the intrinsic
conformal structure of Σ. If n≥4, this conformal
structure determines an intrinsic version of each of the constructions
and results from Section 3.2 in the usual way. (We treat
the n=3 case in Section 3.4, below.) Let denote the
intrinsic conformal metric, and let denote
the intrinsic tractor bundle and its connection on Σ. (In fact, we
shall usually write rather than .) The
conformally invariant (and -parallel) metric on , the
intrinsic tractor metric, shall be denoted and has signature
(n,1). We identify E[w]|Σ with ¯E[w] in the obvious
way.

Now let a local defining function, t, for Σ, as in
Section 2, be given. Henceforth, we let

(16)

na:=dt/|dt|{g}.

Thus na is now a weight 1 conformally invariant conormal field
for Σ and, more generally, for the Σt foliation. Let
. This tensor extends the intrinsic
conformal metric of to a neighbourhood of Σ; its
restriction gives the conformal metric on each leaf Σt0.
Given \textslg∈c, we again define Lab, H, and
on a neighbourhood of Σ as in Section 3.1,
above. This time, however, we replace with , and we use
(16) to define na. As a consequence, Lab, H,
and will have weights 1, −1, and 1, respectively.
This convention will hold for the remainder of this paper. In fact,
all formulae will now carry conformal weights, unless we note
otherwise. This use of weights simplifies the transformation of the
formulae under conformal rescaling. The formulae and results from
Section 3.1 carry over to the present setting in the obvious way.

These remarks apply to natural differential operators, of course, and
this leads to the following proposition.

Proposition 3.2.

Let a pair (w1,w2)∈R2 and natural differential operators
P:TΦ[−w1]→TΦ[−w2] and
P′:TΦ→TΦ be given. Finally, suppose that
for all \textslg∈c. Then P′ is
conformally covariant of bidegree (w1,w2) if and only if P is
conformally invariant. A similar statement holds for operators
mapping TΦ[−w1] to TΦ[−w2]∣∣Σ and
TΦ to TΦ∣∣Σ.

Proof.

Note that (ξˆ\textslg)w=e−wΥ(ξ\textslg)w for any
w∈R. The result thus follows from an elementary argument.
∎

Remark 3.3.

Proposition 3.2 allows us to identify conformally
covariant and conformally invariant operators. In the main new
operator constructions of this paper, we will work with conformally
invariant operators, so for the rest of this paper, we will usually
replace property (2) of Problem 1.2 with the following
equivalent property: P\textslgw,K:E[w]→¯E[w−K] is
conformally invariant. Here K is as in Problem 1.2.

3.4. Tractors and conformal Gauss theory

We will
subsequently exploit a conformally invariant replacement for the Gauss
formula (6). Here we develop this machinery. In the following, we
work along Σ, but the discussion applies to any leaf of the
foliation without adjustment. Thus all quantities defined are
extended into a neighbourhood of Σ. We assume n≥3, except
as noted.

An elementary computation shows that

(17)

whence

(18)

ˆH=H+Υana.

(Since H and na now carry weights, (18) differs
from the conformal transformation law for H that we discussed in
Section 1, above.) It follows from (17) that
is conformally invariant. From (18) and
(9) it follows that

(19)

NA=nbZAb−HXA

is conformally invariant along Σ and, more generally, along each
leaf of the foliation Σt. This is the normal tractor
as defined in [3]. It has conformal weight 0. From
{g}abnanb=1, it follows that hABNANB=1. Let T be
the (conformally invariant) subbundle of TA|Σ whose fibre is
the orthogonal complement (with respect to h) of NA. If n≥4, then exists and has the same rank as T. This suggests
the following proposition. This proposition follows
[43, 57], which in turn follow an equ